To find the value of \( b \) in the quadratic equation \( x^2 + bx - 55 = 0 \) so that the solutions are \( -11 \) and \( 5 \), we can use Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots.
According to Vieta's formulas, if \( r_1 \) and \( r_2 \) are the roots of the quadratic equation \( x^2 + bx + c = 0 \), then:
- The sum of the roots \( r_1 + r_2 = -b \)
- The product of the roots \( r_1 \times r_2 = c \)
Here, we have:
- \( r_1 = -11 \)
- \( r_2 = 5 \)
- The constant term \( c = -55 \)
First, we verify the product of the roots: \[ r_1 \times r_2 = -11 \times 5 = -55 \] This matches with the given constant term, confirming that the roots are correct.
Now, we compute the sum of the roots: \[ r_1 + r_2 = -11 + 5 = -6 \]
Using Vieta's first formula: \[ -(b) = -6 \implies b = 6 \]
Thus, the value of \( b \) must be \( 6 \).
The correct answer is \( \boxed{6} \).